3.3013 \(\int \frac{\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^3} \, dx\)
Optimal. Leaf size=386 \[ -\frac{f (a+b x)^{4/3} (c+d x)^{2/3}}{2 (e+f x)^2 (b e-a f) (d e-c f)}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (-2 a d f-b c f+3 b d e)}{3 (e+f x) (b e-a f) (d e-c f)^2}-\frac{(b c-a d) \log (e+f x) (-2 a d f-b c f+3 b d e)}{18 (b e-a f)^{5/3} (d e-c f)^{7/3}}+\frac{(b c-a d) (-2 a d f-b c f+3 b d e) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{6 (b e-a f)^{5/3} (d e-c f)^{7/3}}+\frac{(b c-a d) (-2 a d f-b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} (b e-a f)^{5/3} (d e-c f)^{7/3}} \]
[Out]
-(f*(a + b*x)^(4/3)*(c + d*x)^(2/3))/(2*(b*e - a*f)*(d*e - c*f)*(e + f*x)^2) + ((3*b*d*e - b*c*f - 2*a*d*f)*(a
+ b*x)^(1/3)*(c + d*x)^(2/3))/(3*(b*e - a*f)*(d*e - c*f)^2*(e + f*x)) + ((b*c - a*d)*(3*b*d*e - b*c*f - 2*a*d
*f)*ArcTan[1/Sqrt[3] + (2*(b*e - a*f)^(1/3)*(c + d*x)^(1/3))/(Sqrt[3]*(d*e - c*f)^(1/3)*(a + b*x)^(1/3))])/(3*
Sqrt[3]*(b*e - a*f)^(5/3)*(d*e - c*f)^(7/3)) - ((b*c - a*d)*(3*b*d*e - b*c*f - 2*a*d*f)*Log[e + f*x])/(18*(b*e
- a*f)^(5/3)*(d*e - c*f)^(7/3)) + ((b*c - a*d)*(3*b*d*e - b*c*f - 2*a*d*f)*Log[-(a + b*x)^(1/3) + ((b*e - a*f
)^(1/3)*(c + d*x)^(1/3))/(d*e - c*f)^(1/3)])/(6*(b*e - a*f)^(5/3)*(d*e - c*f)^(7/3))
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Rubi [A] time = 0.186972, antiderivative size = 386, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used =
{96, 94, 91} \[ -\frac{f (a+b x)^{4/3} (c+d x)^{2/3}}{2 (e+f x)^2 (b e-a f) (d e-c f)}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (-2 a d f-b c f+3 b d e)}{3 (e+f x) (b e-a f) (d e-c f)^2}-\frac{(b c-a d) \log (e+f x) (-2 a d f-b c f+3 b d e)}{18 (b e-a f)^{5/3} (d e-c f)^{7/3}}+\frac{(b c-a d) (-2 a d f-b c f+3 b d e) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{6 (b e-a f)^{5/3} (d e-c f)^{7/3}}+\frac{(b c-a d) (-2 a d f-b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} (b e-a f)^{5/3} (d e-c f)^{7/3}} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^(1/3)/((c + d*x)^(1/3)*(e + f*x)^3),x]
[Out]
-(f*(a + b*x)^(4/3)*(c + d*x)^(2/3))/(2*(b*e - a*f)*(d*e - c*f)*(e + f*x)^2) + ((3*b*d*e - b*c*f - 2*a*d*f)*(a
+ b*x)^(1/3)*(c + d*x)^(2/3))/(3*(b*e - a*f)*(d*e - c*f)^2*(e + f*x)) + ((b*c - a*d)*(3*b*d*e - b*c*f - 2*a*d
*f)*ArcTan[1/Sqrt[3] + (2*(b*e - a*f)^(1/3)*(c + d*x)^(1/3))/(Sqrt[3]*(d*e - c*f)^(1/3)*(a + b*x)^(1/3))])/(3*
Sqrt[3]*(b*e - a*f)^(5/3)*(d*e - c*f)^(7/3)) - ((b*c - a*d)*(3*b*d*e - b*c*f - 2*a*d*f)*Log[e + f*x])/(18*(b*e
- a*f)^(5/3)*(d*e - c*f)^(7/3)) + ((b*c - a*d)*(3*b*d*e - b*c*f - 2*a*d*f)*Log[-(a + b*x)^(1/3) + ((b*e - a*f
)^(1/3)*(c + d*x)^(1/3))/(d*e - c*f)^(1/3)])/(6*(b*e - a*f)^(5/3)*(d*e - c*f)^(7/3))
Rule 96
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])
Rule 94
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && !(SumSimplerQ[p, 1] && !SumSimplerQ[m, 1])
Rule 91
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.)*(x_))), x_Symbol] :> With[{q = Rt[
(d*e - c*f)/(b*e - a*f), 3]}, -Simp[(Sqrt[3]*q*ArcTan[1/Sqrt[3] + (2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/
3))])/(d*e - c*f), x] + (Simp[(q*Log[e + f*x])/(2*(d*e - c*f)), x] - Simp[(3*q*Log[q*(a + b*x)^(1/3) - (c + d*
x)^(1/3)])/(2*(d*e - c*f)), x])] /; FreeQ[{a, b, c, d, e, f}, x]
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^3} \, dx &=-\frac{f (a+b x)^{4/3} (c+d x)^{2/3}}{2 (b e-a f) (d e-c f) (e+f x)^2}+\frac{(3 b d e-b c f-2 a d f) \int \frac{\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx}{3 (b e-a f) (d e-c f)}\\ &=-\frac{f (a+b x)^{4/3} (c+d x)^{2/3}}{2 (b e-a f) (d e-c f) (e+f x)^2}+\frac{(3 b d e-b c f-2 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 (b e-a f) (d e-c f)^2 (e+f x)}-\frac{((b c-a d) (3 b d e-b c f-2 a d f)) \int \frac{1}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)} \, dx}{9 (b e-a f) (d e-c f)^2}\\ &=-\frac{f (a+b x)^{4/3} (c+d x)^{2/3}}{2 (b e-a f) (d e-c f) (e+f x)^2}+\frac{(3 b d e-b c f-2 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 (b e-a f) (d e-c f)^2 (e+f x)}+\frac{(b c-a d) (3 b d e-b c f-2 a d f) \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{3 \sqrt{3} (b e-a f)^{5/3} (d e-c f)^{7/3}}-\frac{(b c-a d) (3 b d e-b c f-2 a d f) \log (e+f x)}{18 (b e-a f)^{5/3} (d e-c f)^{7/3}}+\frac{(b c-a d) (3 b d e-b c f-2 a d f) \log \left (-\sqrt [3]{a+b x}+\frac{\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{6 (b e-a f)^{5/3} (d e-c f)^{7/3}}\\ \end{align*}
Mathematica [C] time = 0.234971, size = 175, normalized size = 0.45 \[ \frac{\sqrt [3]{a+b x} \left (\frac{2 (e+f x) (-2 a d f-b c f+3 b d e) \left ((c+d x) (b e-a f)-(e+f x) (b c-a d) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )}{(b e-a f) (d e-c f)}-3 f (a+b x) (c+d x)\right )}{6 \sqrt [3]{c+d x} (e+f x)^2 (b e-a f) (d e-c f)} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^(1/3)/((c + d*x)^(1/3)*(e + f*x)^3),x]
[Out]
((a + b*x)^(1/3)*(-3*f*(a + b*x)*(c + d*x) + (2*(3*b*d*e - b*c*f - 2*a*d*f)*(e + f*x)*((b*e - a*f)*(c + d*x) -
(b*c - a*d)*(e + f*x)*Hypergeometric2F1[1/3, 1, 4/3, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]))/((b*e
- a*f)*(d*e - c*f))))/(6*(b*e - a*f)*(d*e - c*f)*(c + d*x)^(1/3)*(e + f*x)^2)
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Maple [F] time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( fx+e \right ) ^{3}}\sqrt [3]{bx+a}{\frac{1}{\sqrt [3]{dx+c}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^3,x)
[Out]
int((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^3,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{1}{3}}}{{\left (d x + c\right )}^{\frac{1}{3}}{\left (f x + e\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^3,x, algorithm="maxima")
[Out]
integrate((b*x + a)^(1/3)/((d*x + c)^(1/3)*(f*x + e)^3), x)
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Fricas [B] time = 9.88236, size = 9846, normalized size = 25.51 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^3,x, algorithm="fricas")
[Out]
[1/18*(3*sqrt(1/3)*(3*(b^3*c*d^2 - a*b^2*d^3)*e^5 - (4*b^3*c^2*d + a*b^2*c*d^2 - 5*a^2*b*d^3)*e^4*f + (b^3*c^3
+ 5*a*b^2*c^2*d - 4*a^2*b*c*d^2 - 2*a^3*d^3)*e^3*f^2 - (a*b^2*c^3 + a^2*b*c^2*d - 2*a^3*c*d^2)*e^2*f^3 + (3*(
b^3*c*d^2 - a*b^2*d^3)*e^3*f^2 - (4*b^3*c^2*d + a*b^2*c*d^2 - 5*a^2*b*d^3)*e^2*f^3 + (b^3*c^3 + 5*a*b^2*c^2*d
- 4*a^2*b*c*d^2 - 2*a^3*d^3)*e*f^4 - (a*b^2*c^3 + a^2*b*c^2*d - 2*a^3*c*d^2)*f^5)*x^2 + 2*(3*(b^3*c*d^2 - a*b^
2*d^3)*e^4*f - (4*b^3*c^2*d + a*b^2*c*d^2 - 5*a^2*b*d^3)*e^3*f^2 + (b^3*c^3 + 5*a*b^2*c^2*d - 4*a^2*b*c*d^2 -
2*a^3*d^3)*e^2*f^3 - (a*b^2*c^3 + a^2*b*c^2*d - 2*a^3*c*d^2)*e*f^4)*x)*sqrt(-(b^2*d*e^3 - a^2*c*f^3 - (b^2*c +
2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)^(1/3)/(d*e - c*f))*log(-(3*a^2*c*f^2 + (b^2*c + 2*a*b*d)*e^2 - 2*(2
*a*b*c + a^2*d)*e*f - 3*(b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)^(1/3)*(b*e
- a*f)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + (3*b^2*d*e^2 - 2*(b^2*c + 2*a*b*d)*e*f + (2*a*b*c + a^2*d)*f^2)*x -
3*sqrt(1/3)*(2*(b*d*e^2 + a*c*f^2 - (b*c + a*d)*e*f)*(b*x + a)^(2/3)*(d*x + c)^(1/3) - (b^2*d*e^3 - a^2*c*f^3
- (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) - (b^2*d*e^3 - a^2*
c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)^(1/3)*(b*c*e - a*c*f + (b*d*e - a*d*f)*x))*sqrt(-(b
^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)^(1/3)/(d*e - c*f)))/(f*x + e)) - (b^
2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)^(2/3)*(3*(b^2*c*d - a*b*d^2)*e^3 - (b
^2*c^2 + a*b*c*d - 2*a^2*d^2)*e^2*f + (3*(b^2*c*d - a*b*d^2)*e*f^2 - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*f^3)*x^2
+ 2*(3*(b^2*c*d - a*b*d^2)*e^2*f - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*e*f^2)*x)*log(((b*d*e^2 + a*c*f^2 - (b*c +
a*d)*e*f)*(b*x + a)^(2/3)*(d*x + c)^(1/3) + (b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*
d)*e*f^2)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + (b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c
+ a^2*d)*e*f^2)^(1/3)*(b*c*e - a*c*f + (b*d*e - a*d*f)*x))/(d*x + c)) + 2*(b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*
a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)^(2/3)*(3*(b^2*c*d - a*b*d^2)*e^3 - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*e^2
*f + (3*(b^2*c*d - a*b*d^2)*e*f^2 - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*f^3)*x^2 + 2*(3*(b^2*c*d - a*b*d^2)*e^2*f
- (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*e*f^2)*x)*log(((b*d*e^2 + a*c*f^2 - (b*c + a*d)*e*f)*(b*x + a)^(1/3)*(d*x +
c)^(2/3) - (b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)^(2/3)*(d*x + c))/(d*x +
c)) + 3*(6*b^3*d^2*e^5 - 3*a^3*c^2*f^5 - (8*b^3*c*d + 19*a*b^2*d^2)*e^4*f + 2*(b^3*c^2 + 13*a*b^2*c*d + 10*a^
2*b*d^2)*e^3*f^2 - 7*(a*b^2*c^2 + 4*a^2*b*c*d + a^3*d^2)*e^2*f^3 + 2*(4*a^2*b*c^2 + 5*a^3*c*d)*e*f^4 + (3*b^3*
d^2*e^4*f - 2*(b^3*c*d + 5*a*b^2*d^2)*e^3*f^2 - (b^3*c^2 - 8*a*b^2*c*d - 11*a^2*b*d^2)*e^2*f^3 + 2*(a*b^2*c^2
- 5*a^2*b*c*d - 2*a^3*d^2)*e*f^4 - (a^2*b*c^2 - 4*a^3*c*d)*f^5)*x)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(b^3*d^3*e
^8 + a^3*c^3*e^2*f^6 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^7*f + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*e^6*f^2 - (
b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*e^5*f^3 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*e^4*f^4
- 3*(a^2*b*c^3 + a^3*c^2*d)*e^3*f^5 + (b^3*d^3*e^6*f^2 + a^3*c^3*f^8 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^5*f^3 + 3*
(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*e^4*f^4 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*e^3*f^5
+ 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*e^2*f^6 - 3*(a^2*b*c^3 + a^3*c^2*d)*e*f^7)*x^2 + 2*(b^3*d^3*e^7*f
+ a^3*c^3*e*f^7 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^6*f^2 + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*e^5*f^3 - (b^3
*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*e^4*f^4 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*e^3*f^5 -
3*(a^2*b*c^3 + a^3*c^2*d)*e^2*f^6)*x), -1/18*(6*sqrt(1/3)*(3*(b^3*c*d^2 - a*b^2*d^3)*e^5 - (4*b^3*c^2*d + a*b^
2*c*d^2 - 5*a^2*b*d^3)*e^4*f + (b^3*c^3 + 5*a*b^2*c^2*d - 4*a^2*b*c*d^2 - 2*a^3*d^3)*e^3*f^2 - (a*b^2*c^3 + a^
2*b*c^2*d - 2*a^3*c*d^2)*e^2*f^3 + (3*(b^3*c*d^2 - a*b^2*d^3)*e^3*f^2 - (4*b^3*c^2*d + a*b^2*c*d^2 - 5*a^2*b*d
^3)*e^2*f^3 + (b^3*c^3 + 5*a*b^2*c^2*d - 4*a^2*b*c*d^2 - 2*a^3*d^3)*e*f^4 - (a*b^2*c^3 + a^2*b*c^2*d - 2*a^3*c
*d^2)*f^5)*x^2 + 2*(3*(b^3*c*d^2 - a*b^2*d^3)*e^4*f - (4*b^3*c^2*d + a*b^2*c*d^2 - 5*a^2*b*d^3)*e^3*f^2 + (b^3
*c^3 + 5*a*b^2*c^2*d - 4*a^2*b*c*d^2 - 2*a^3*d^3)*e^2*f^3 - (a*b^2*c^3 + a^2*b*c^2*d - 2*a^3*c*d^2)*e*f^4)*x)*
sqrt((b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)^(1/3)/(d*e - c*f))*arctan(sqr
t(1/3)*(2*(b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)^(2/3)*(b*x + a)^(1/3)*(d
*x + c)^(2/3) + (b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)^(1/3)*(b*c*e - a*c
*f + (b*d*e - a*d*f)*x))*sqrt((b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)^(1/3
)/(d*e - c*f))/(b^2*c*e^2 - 2*a*b*c*e*f + a^2*c*f^2 + (b^2*d*e^2 - 2*a*b*d*e*f + a^2*d*f^2)*x)) + (b^2*d*e^3 -
a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)^(2/3)*(3*(b^2*c*d - a*b*d^2)*e^3 - (b^2*c^2 +
a*b*c*d - 2*a^2*d^2)*e^2*f + (3*(b^2*c*d - a*b*d^2)*e*f^2 - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*f^3)*x^2 + 2*(3*(b
^2*c*d - a*b*d^2)*e^2*f - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*e*f^2)*x)*log(((b*d*e^2 + a*c*f^2 - (b*c + a*d)*e*f)
*(b*x + a)^(2/3)*(d*x + c)^(1/3) + (b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)
^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + (b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*
e*f^2)^(1/3)*(b*c*e - a*c*f + (b*d*e - a*d*f)*x))/(d*x + c)) - 2*(b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^
2*f + (2*a*b*c + a^2*d)*e*f^2)^(2/3)*(3*(b^2*c*d - a*b*d^2)*e^3 - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*e^2*f + (3*(
b^2*c*d - a*b*d^2)*e*f^2 - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*f^3)*x^2 + 2*(3*(b^2*c*d - a*b*d^2)*e^2*f - (b^2*c^
2 + a*b*c*d - 2*a^2*d^2)*e*f^2)*x)*log(((b*d*e^2 + a*c*f^2 - (b*c + a*d)*e*f)*(b*x + a)^(1/3)*(d*x + c)^(2/3)
- (b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)^(2/3)*(d*x + c))/(d*x + c)) - 3*
(6*b^3*d^2*e^5 - 3*a^3*c^2*f^5 - (8*b^3*c*d + 19*a*b^2*d^2)*e^4*f + 2*(b^3*c^2 + 13*a*b^2*c*d + 10*a^2*b*d^2)*
e^3*f^2 - 7*(a*b^2*c^2 + 4*a^2*b*c*d + a^3*d^2)*e^2*f^3 + 2*(4*a^2*b*c^2 + 5*a^3*c*d)*e*f^4 + (3*b^3*d^2*e^4*f
- 2*(b^3*c*d + 5*a*b^2*d^2)*e^3*f^2 - (b^3*c^2 - 8*a*b^2*c*d - 11*a^2*b*d^2)*e^2*f^3 + 2*(a*b^2*c^2 - 5*a^2*b
*c*d - 2*a^3*d^2)*e*f^4 - (a^2*b*c^2 - 4*a^3*c*d)*f^5)*x)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(b^3*d^3*e^8 + a^3*
c^3*e^2*f^6 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^7*f + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*e^6*f^2 - (b^3*c^3 +
9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*e^5*f^3 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*e^4*f^4 - 3*(a^2
*b*c^3 + a^3*c^2*d)*e^3*f^5 + (b^3*d^3*e^6*f^2 + a^3*c^3*f^8 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^5*f^3 + 3*(b^3*c^2*
d + 3*a*b^2*c*d^2 + a^2*b*d^3)*e^4*f^4 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*e^3*f^5 + 3*(a*b^
2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*e^2*f^6 - 3*(a^2*b*c^3 + a^3*c^2*d)*e*f^7)*x^2 + 2*(b^3*d^3*e^7*f + a^3*c^3
*e*f^7 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^6*f^2 + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*e^5*f^3 - (b^3*c^3 + 9*
a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*e^4*f^4 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*e^3*f^5 - 3*(a^2*b*
c^3 + a^3*c^2*d)*e^2*f^6)*x)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [3]{a + b x}}{\sqrt [3]{c + d x} \left (e + f x\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**(1/3)/(d*x+c)**(1/3)/(f*x+e)**3,x)
[Out]
Integral((a + b*x)**(1/3)/((c + d*x)**(1/3)*(e + f*x)**3), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{1}{3}}}{{\left (d x + c\right )}^{\frac{1}{3}}{\left (f x + e\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^3,x, algorithm="giac")
[Out]
integrate((b*x + a)^(1/3)/((d*x + c)^(1/3)*(f*x + e)^3), x)